Prove for any integer n if n31 is even then n is odd Prove f

Prove: for any integer n, if (n^3)+1 is even then n is odd

Prove: for any integer n, (n^2)+4 is not divisible by 3

Consider n an odd number

Obviously n^3 is odd and hence n^3+1 is even.

Let n be any integer

n^2 will be of the form n x n

There are 3 possibilities for n

Either n gives remainder 0 when divided by 3, or 1 or 2

Case I: o remainder

Then n^2+4 = (3m)^2+3+1 gives remainder 1 when divided by 3. Hence true.

CAse 2: 1 remainder

n = 3m+1

n^2+4 = 9m^2+6m+5 gives remainder 2

Case 3: 2 remainder

n = 3m+2

n^2+4 = 9m^2+12m+8 gives again remaider 2

Hence true for all n.